Question

Suppose $X^{~}N(15,3)$. Between what $x$values does $68.27\%$of the data lie? The range of $x$values is centered at the mean of the distribution (i.e., 15).

Short answer

The $68.27\%$of the data lie between $(12,18)$.

Step-by-step solution

Given information

Given in the question that, $X^{~}N(15,3)$

We have to find between what $x$values does $68.27\%$of the data lie

Explanation 

According to the empirical rule of normal distribution, approximately $68\%$values $x$lies within one standard deviation of the mean.

Therefore, $68.27\%$of data also will lie approximately near to within one standard deviation of the mean.

From the information given in the question, the random variable $x$follows normal distribution with parameter $(15,3)$

Hence, the range of $x$values can be compute as follow:

$=\mu \pm l\sigma$

$=15\pm 1\times 3$

$=15\pm 3$

$=(12,18)$